{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:W3D5K7GAA7QXDF5SW4QKBR3VAY","short_pith_number":"pith:W3D5K7GA","schema_version":"1.0","canonical_sha256":"b6c7d57cc007e17197b2b720a0c775060336f9edea1060da03a47a1ebdc89182","source":{"kind":"arxiv","id":"2412.00635","version":2},"attestation_state":"computed","paper":{"title":"Critical threshold for regular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Ishaan Bhadoo","submitted_at":"2024-12-01T01:22:47Z","abstract_excerpt":"In this article, we study the critical percolation threshold $p_c$ for $d$-regular graphs. It is well-known that $p_c \\geq \\frac{1}{d-1}$ for such graphs, with equality holding for the $d$-regular tree. We prove that among all quasi-transitive $d$-regular graphs, the equality $p_c(G) = \\frac{1}{d-1}$ holds if and only if $G$ is a tree. Furthermore, we provide counterexamples that illustrate the necessity of the quasi-transitive assumption."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2412.00635","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-12-01T01:22:47Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"823e6b0a72216c11098c1ca05d2786b12373b964f0cd7cafb60149e9b34947a0","abstract_canon_sha256":"658a9ce26c785b61e24fc1d9256ac2d823886095a80d18a1254e28372bcc3c11"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:59:01.207214Z","signature_b64":"VYUO5Hl+1WYi0j0HxNHz6dnEZs5c8kvWjjZcgfArvUexkJBITlCgw+A0wzAHICC1K6gkqCapC18OwrXrT1LTAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b6c7d57cc007e17197b2b720a0c775060336f9edea1060da03a47a1ebdc89182","last_reissued_at":"2026-07-05T09:59:01.206770Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:59:01.206770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Critical threshold for regular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Ishaan Bhadoo","submitted_at":"2024-12-01T01:22:47Z","abstract_excerpt":"In this article, we study the critical percolation threshold $p_c$ for $d$-regular graphs. It is well-known that $p_c \\geq \\frac{1}{d-1}$ for such graphs, with equality holding for the $d$-regular tree. We prove that among all quasi-transitive $d$-regular graphs, the equality $p_c(G) = \\frac{1}{d-1}$ holds if and only if $G$ is a tree. Furthermore, we provide counterexamples that illustrate the necessity of the quasi-transitive assumption."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.00635","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.00635/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2412.00635","created_at":"2026-07-05T09:59:01.206824+00:00"},{"alias_kind":"arxiv_version","alias_value":"2412.00635v2","created_at":"2026-07-05T09:59:01.206824+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.00635","created_at":"2026-07-05T09:59:01.206824+00:00"},{"alias_kind":"pith_short_12","alias_value":"W3D5K7GAA7QX","created_at":"2026-07-05T09:59:01.206824+00:00"},{"alias_kind":"pith_short_16","alias_value":"W3D5K7GAA7QXDF5S","created_at":"2026-07-05T09:59:01.206824+00:00"},{"alias_kind":"pith_short_8","alias_value":"W3D5K7GA","created_at":"2026-07-05T09:59:01.206824+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY","json":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY.json","graph_json":"https://pith.science/api/pith-number/W3D5K7GAA7QXDF5SW4QKBR3VAY/graph.json","events_json":"https://pith.science/api/pith-number/W3D5K7GAA7QXDF5SW4QKBR3VAY/events.json","paper":"https://pith.science/paper/W3D5K7GA"},"agent_actions":{"view_html":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY","download_json":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY.json","view_paper":"https://pith.science/paper/W3D5K7GA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2412.00635&json=true","fetch_graph":"https://pith.science/api/pith-number/W3D5K7GAA7QXDF5SW4QKBR3VAY/graph.json","fetch_events":"https://pith.science/api/pith-number/W3D5K7GAA7QXDF5SW4QKBR3VAY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY/action/storage_attestation","attest_author":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY/action/author_attestation","sign_citation":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY/action/citation_signature","submit_replication":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY/action/replication_record"}},"created_at":"2026-07-05T09:59:01.206824+00:00","updated_at":"2026-07-05T09:59:01.206824+00:00"}